1. Lab 4 Application of RTOS
11. Use FFT to Design DF
Lab 4 Application of RTOS
Input sound, analog filter
Digital filter, FFT
Display amplitude versus freq on the oLED
Reference EE345L book, chapter 15
July 5, 2018
Jonathan Valvano EE345M/EE380L.6

2. Jonathan Valvano EE345M/EE380L.6
Lab4 Spectrum Analyzer
July 5, 2018
Jonathan Valvano EE345M/EE380L.6

3. Design a linear FIR digital filter
Specify gain versus frequency
Specify phase versus frequency
y(n)
x(n)
July 5, 2018
Jonathan Valvano EE345M/EE380L.6

4. Fast Fourier Transform (FFT)
FFT is a faster version of the Discrete Fourier Transform (DFT)
FFT spectrum of a cosine, which has a frequency of 0.1 Fs
Fs = Hz
Cosine freq = 1000 Hz
Region interested in from 0 to Fs/2 (symmetric about Fs/2)
July 5, 2018
Jonathan Valvano EE345M/EE380L.6

5. Jonathan Valvano EE345M/EE380L.6
FIR digital filter
Y(z) = H(z) X(z)
h(n) = IFFT {H(z)}
Convolution
y(n) = h(n)*x(n)
Constants h0, h1, ... hN-1
y(n) = h0·x(n)+h1·x(n-1) hN-1·x(n-(N-1))
N multiplies, N-1 additions per sample
July 5, 2018
Jonathan Valvano EE345M/EE380L.6

6. How to choose sampling rate
Nyquist Rate
Limitation of display
Limitation of processor
Limitation of RAM
Limitation of human eyes and ears
Limitation of communication channel
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Jonathan Valvano EE345M/EE380L.6

7. How to choose number of samples
Frequency resolution = fs/N
Increase in N results in better frequency resolution
However, increase in N leads to a bigger MACQ buffer and more multiplies and additions
Does not need to be a power of 2
DFT calculated once, off line
July 5, 2018
Jonathan Valvano EE345M/EE380L.6

8. Jonathan Valvano EE345M/EE380L.6
Design process (1)
Specify desired gain and phase, 0 to ½ fs
k goes from 0 to N/2 (f = k fs/N)
H(k) is complex
|H(k)| is gain
angle(H(k)) is phase
For ½ fs to fs
H(N-k) is complex conjugate of H(k)
Poles and zeros are in complex conjugate pairs
July 5, 2018
Jonathan Valvano EE345M/EE380L.6

9. Jonathan Valvano EE345M/EE380L.6
Design process (2)
Take IDFT of H(k) to yield h(n)
n goes from 0 to N-1
h(n) will be real, because H(k) symmetric
The digital filter is
y(n) = h0·x(n)+h1·x(n-1) hN-1·x(n-(N-1))
July 5, 2018
Jonathan Valvano EE345M/EE380L.6

10. Binary Fixed point notation
Q formats
Qn Number (16 bit)
n: specifies no of bits used for precision
16-n: no of bits used for Range
Eg: (unsigned number) - What Qn?
How is this number stored as an integer?
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Jonathan Valvano EE345M/EE380L.6

11. Example Open FIRdesign51.xls Change sampling rate to 10,000 Hz
Adjust red desired gain to make BPF
Pass 2 to 4 kHz
Look at sharp corner versus round corner
Notice linear phase
Copy 51 coefficients into software

12. 2k to 4k BPF

13. FIR Filter SW design short Filter(short data){unsigned int k;
const long h[51]={-3,-9,4,5,0,17,5,-20,-5,-7,-22,
24,41,-8,2,1,-74,-31,71,20,33,125,-119,-350,67,
462,67,-350,-119,125,33,20,71,-31,-74,1,2,-8,41,
24,-22,-7,-5,-20,5,17,0,5,4,-9,-3};
static unsigned int n=50; // 51,52,… 101
short Filter(short data){unsigned int k;
static short x[102]; // this MACQ needs twice
short y;
n++;
if(n==102) n=51;
x[n] = x[n-51] = data; // two copies of new data
y = 0;
for(k=0;k<51;k++){
y = y + h[k]*x[n-k]; // convolution }
y = y/256; // fixed point
return y;

14. Jonathan Valvano EE345M/EE380L.6
Circular Buffering
Source: “Communication system design using DSP algorithms” by Steven A. Tretter (chapter 3, page 73)
July 5, 2018
Jonathan Valvano EE345M/EE380L.6

15. Optimization Pointer implementations of MACQ faster
Do not try and shift the data
Convolution x[n]*h[n] takes N multiplies, N-1 additions per sample
Can be optimized to N/2 multiplies
Coefficients are symmetric
Assembly optimization with MLA
Multiply with accumulate